I am currently on season 3 of Star Trek: TOS, and wanted to know if/how watching in this order will affect the story (NO SPOILERS PLEASE!!!):
Star Trek: TOS
Star Trek: TAS
Star Trek movies 1-7
Star Trek: TNG
Star Trek movies 8-10
Star Trek: DS9
Star Trek: VOY
Star Trek: ENT
If there is a problem please say how I should watch instead. Thanks!
I suggest you watch Star Trek: TNG before watching the seventh movie (Star Trek: Generations). Generations is far more of a TNG movie than a TOS movie.
To make things more interesting, the first season of Star Trek: DS9 aired the same year as the seventh (last) season of Star Trek: TNG - so there is some overlap there (and crossing over) there to get the new series started. You might want to interleave those if you're really worried about spoilers.
TNG and DS9 seasons that are at the same time need to be interleaved because there is crossovers so it's best to just interleave them. All the other seasons that overlap don't cross over to my recollection so you don't need to worry about it... It is probably best to watch the rest of DS9's seasons straight through due to the serial nature of the show.
A good way to watch Star Trek is along with the podcast Mission Log. They're only on TNG and will take several more years to get through the rest, but I'd definitely recommend it as a way to watch ST thus far http://missionlogpodcast.com
The problem with out-of-order viewing isn't so much spoilers as coherence. By and large, things will make the most sense if viewed in production order (not necessarily the same as broadcast/release order). For example, TOS "The Cage" could be the ideal place to start because it was the very first episode written/produced, even though it was never aired as such (but was cut into the later two-part "The Menagerie"), and only presented in its original form decades later. Likewise, much of what eventually became the Star Trek canon as we know it evolved (in production order) over the first 8 or 10 episodes of the first TOS season. Watching them in broadcast order (as everyone would have back in 1966) might seem a bit confused; sets, costumes (uniforms), Spock's look and demeanor, terminology, etc. all evolved and took until about the 10th episode to crystallize.
TNG had some minor character development over its run (one major cast change approaching the end of the first season), and some stories reference events from earlier episodes, so would make the most sense to watch in order. Also, the very first and very last episodes (both 2-parters) bookend the series, so should be watched in series order.
DS9 had major arcs in its later seasons, and also had bookending first and last episodes.
Voyager can be watched out of order, excepting for the first episode which establishes the show and the last episode which, again serves as a bookend. There are no story arcs, however there are several character arcs which will make more sense when the show is watched in broadcast order, especially a major casting change which occurs in the two-part cliffhanger bridging seasons 3&4.
Enterprise's third season has a major arc, and should be watched in order. Again, the very first and last episodes of the series are bookends. Watching the entire series in order can offer one reward - seeing the familiar Trek technology and other conventions build up a bit at a time.
As for the movies: #3 (Search for Spock) builds its story on the ending of #2 (Wrath of Khan). Although #4 (Voyage Home) tells a stand-alone story, its beginning starts with the state at which #3 ends. This makes movies 2,3, and 4 a sort of loose trilogy.
If you're the sort of person who believes that The Magician's Nephew is the first book in "The Chronicles of Narnia," then maybe you should watch "Star Trek: ENT" first, because it is set in an earlier time (before "Star Trek: TOS").
